m with ai rnn and bi rn consider the problem of

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Unformatted text preview: ebsite and put it in your Matlab path before the default version (which has a bug). 7.14 Isoperimetric problem. We consider the problem of choosing a curve in a two-dimensional plane that encloses as much area as possible between itself and the x-axis, subject to constraints. For simplicity we will consider only curves of the form C = {(x, y ) | y = f (x)}, where f : [0, a] → R. This assumes that for each x-value, there can only be a single y -value, which need not be the case for general curves. We require that at the end points (which are given), the 69 curve returns to the x-axis, so f (0) = 0, and f (a) = 0. In addition, the length of the curve cannot exceed a budget L, so we must have a 0 1 + f ′ (x)2 dx ≤ L. The objective is the area enclosed, which is given by a f (x) dx. 0 To pose this as a finite dimensional optimization problem, we discretize over the x-values. Specifically, we take xi = h(i − 1), i = 1, . . . , N + 1, where h = a/N is the discretization step size, and we let yi = f (xi ). Thus our objective becomes N yi , h i=1 and our constra...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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